study on ultrasonic stepped horn geometry - Proiecte POS-DRU

Nonconventional Technologies Review – no. 4/2011
STUDY ON ULTRASONIC STEPPED HORN GEOMETRY
DESIGN AND FEM SIMULATION
Eng. ALEXANDRU SERGIU NANU1, Prof. Niculae Ion MARINESCU2,
Assoc. Prof. Daniel GHICULESCU3
1
2
Politehnica” University of Bucharest, [email protected], “Politehnica” University of Bucharest,
3
[email protected], ““Politehnica” University of Bucharest, [email protected]
Abstract: The paper deals with study through Finite Element Method (FEM) of ultrasonic stepped horn
used to aid electro-discharge machining (EDM+US). The thermal phenomena are dominant within EDM
machining mechanism. The role of US aiding is to intensify the material thermal removal through cavitational
phenomena.Several studies were carried on in order to analyze the influence of steps lengths and diameters
and corner radius size on own frequency of stepped ultrasonic horns. A good agreement between the FEM
results and theoretical and experimental data was emphasized. Methods to adjust the own frequency of
stepped horns by adjusting the lengths, diameters and additional machining of a groove at gravity centre
level of horn are presented.
Keywords: stepped sonotrode design, ultrasonics, FEM, eigenfrequency, EDM.
1. INTRODUCTION
Significant increasing in performance and
qualitative improvements are achieved by
using ultrasonic vibrations in machining
technological processes. Applications of
ultrasonic vibration energy in machining
technologies are realized by two different
approaches. The first approach, called as an
ultrasonic machining, is based on abrasive
principle of material removal. The tool which
is shaped in the exact configuration to be
ground in workpiece and it is attached to a
vibrating horn. The second approach is based
on the conventional machining technologies–
ultrasonic aided machining.
The ultraacoustic systems composed by
the generator and the ultrasonic converter
(transducer and wave adapting concentrator)
can be built in order to work at a fix frequency
named resonance frequency. The optimal
frequency can be obtained starting from a
lower own frequency close to resonance one
of ultrasonic stepped horn (USH) and
adjusting its dimensions [1].
The usual frequencies for ultrasonic aided
EDM are 20 and 40kHz. In this paper, the first
one was used.
The most frequently used shapes of
ultrasonic horns are: cylindrical, tapered,
exponential and stepped. To achieve optimal
performance of ultrasonic machining system,
is necessary to take into account all relevant
effects and parameters that affect the
dynamics of the system. One of the most
important element of the ultrasonic system –
sonotrode (ultrasonic horn), must have the
required dynamic properties, which must be
determined already in design phase.
Ultrasonic chain (fig. 1) consists in typical
transducer sandwhich piezo-electric ceramics
center-bolt (Langevin) design (the housing
and electrode leads are not shown) and a
stepped ultrasonic horn (USH). Ultrasonic
transducer converts the electrical wave to
mechanical vibration which is relative small
and must be amplified using an acoustic
horn.
Fig. 1. Ultrasonic chain
The primary function of the horn is to
amplify the vibration of the tool to the level
required for effective machining assistance,
but it serves also as a means of transmitting
the vibrational energy from the transducer.
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Nonconventional Technologies Review – no. 4/2011
It does so by being in resonance with the
transducer.
The design and manufacture of the horn
require special attention because an
incorrectly manufactured horn will decrease
machining performance and can lead to the
destruction of the vibration system and cause
significant damage to the generator.
- own frequency of the transducer 20250 Hz
(specified by manufacturer);
- coupling diameter of transducer Ø 52 mm;
- tool-electrode dimensions: Ø 40 mm, height
20mm(resulted from EDM process demands);
- tool-electrode material copper (Cu – 99.5%);
- sonotrode material same as the electrodetool;
- sonotrode type: stepped horn (total horn
length equal with half wavelength – λ/2);
-frequency value of blank sonotrode (lower
than final value) used for dimensional
calculus fcalc= 19750 Hz.
2. DESIGN PROCEDURE OF USH
The critical condition of USH design is
own resonance frequency witch must match
the working frequency of the ultrasonic
transducer [3], [4].
If so, the correct frequency must be
attended. Amplitude of the horn tip
displacement must be known. Last concern is
that high stress occurs in nodal area where
section area varies.
For conception and achievement of
ultrasonic assembly, all the elements that
compose the ultrasonic chain must be
properly sized for that the system to perform
at the resonant frequency. For this is
necessary to cover the following steps:
- frequency selection;
- selection of the proper material;
- determination of the sound propagation
velocity in the selected material;
- calculation of the theoretical dimensions;
- realization of the documentation needed
for execution of the ultrasonic assembly;
-achievement of the prototypes, accords,
and test case of our experimental data.
Fig. 3. Stepped ultrasonic horn
(correlation L0 - λ and amplitudes )
The transformation ratio (gain in
amplitude) can be computed using (see fig.3):
ξ
Kt = 2
ξ1
D 
= 2 
 d1 
2
(1)
where Kt – is the theoretical amplitude gain.
For the present case Kt = 1,69.
The sound velocity measured on a
standard sample made from the same
material as the sonotrode (copper Cu–99,5%)
is cs= 4010 m/s.
According to [2] the junction to the
smaller diameter is on the nodal plane, and
the total length of the accoustic horn L0 can
be calculated using the relation (2):
L0 = L1 + L2 =k1
cs
c
+k 2 s
4f calc
4f calc
(2)
where the correction factors k1 and k2
depends on the given sonotrode crosssections.
Asumming Merkulov şi Kharitonov [1]
theory the length of the stepped horn can be
obtained using the following formulae:
Fig. 2. Stepped ultrasonic horn
(geometry and main dimensions)
The length (L0) of the sonotrode normally
coresponds to a half wavelength (fig.2).
Calculations of sonotrode dimensions are
based on the following input data:
L1 =
26
1,5
ku
and L2 =
1,6
ku
[m]
(3)
Nonconventional Technologies Review – no. 4/2011
where:
and
2π
λ
c
λ= s
f calc
ku =
evaluate the effects of these variations to
understand their impact on the results.
The adequate introduction of data into
the calculus program leads to accuracy
obtaining of the nodal zones and obtaining
the maximum efficiency amplitude factor of
the ultrasonic assembly elements.
(4)
(5)
Using the input data mentioned above
we achieved the following results:
L1=0,04805 m and L2=0,05125 m.
Figure 4 shows all the required
dimensions for sonotrode manufacturing.
4. FEM RESULTS
The COMSOL Multiphysics® “Structural
Mechanics” - Eigenfrequency module was
used to perform the modal analysis.
Defining characteristics of stepped horn
material. In order to construct the sonotrode
we use copper (Cu -99,5%) material whose
technical parameters were introduced (sound
propagation velocity, elongation modulus,
density, Poisson coefficient):
Parameter
Young’s modulus
Density
Poisson’s ratio
unit
GPa
kg/m3
-
value
148.
8800.
0.33
Mesh generation. According to the basic
principles of finite element method theory, the
smaller the mesh element size is, the more
accurate the results of an analysis will be. If
the mesh element size is infinitely small, the
theoretical model will approach the optimal
solution. In the analysis process, when used
elements are too small, the meshing will
generate too many elements, nodes and
deegres of freedom for the model in general.
This increases computational intensity,
resulting a model that is either too timeconsuming to be solved, or potential errors
could occur. Reasonable mesh element size
is a factor that has to be considered in the
present modelling.
Fig. 4. Calculated dimensions of USH
3. FEM MODELLING OF USH
FEM, just as any analytical method, is
used by engineers to model, analyze, and
predict the performance of real physical
systems. The accuracy of the model depends
not so much on the particular analytical
technique, but on the assumptions made in
modelling the physical system. The accuracy
of the model must be validated by
benchmarking its predictions to experimental
data and results from simpler problems.
Results from the computer or any analysis
should never be assumed to be correct just
because one can “turn the crank” on the
problem.
A physical problem is defined by its
geometry, material properties, boundary
conditions, and the external loads, which can
be distributed in time and space. Sometimes
these parameters for a particular problem are
not that well understood. Material properties
are often not accurately known, might have to
be extrapolated from reference values, or
might have a relatively high degree of
variability. In such cases, once the model is
known to be an accurate representation, the
engineer performs parametric studies to
27
461 elements for a 2D 16816 elements for a
axis symetric model
3D model
Fig. 5. Meshing statistics
Nonconventional Technologies Review – no. 4/2011
A good agreement between the results
obtained from COMSOL application and
theoretically computed data has been found
as mentioned below:
f2D axisymetric = 19615 HZ (relative error: -0,68%)
f3D = 19686 Hz (relative error: -0,32%).
The frequency of the blank sonotrode is
the result of additional material (generally 1-2
mm). This determines a frequency 5001000Hz below the desired resonant
frequency. The sonotrode reaches the
requiered frequency after several iterations of
shortening and remeasuring.
The sonotrode frequency measuring
system is used to determine its natural
frequency. The measuring system consists of
a generator, an amplitude transducer and a
frequency indicator. The resonant frequency
has been obtained when the maximum
amplitude is achieved on the reading
instrument. Shortening must be done on both
front surfaces (as seen on fig.10).
Practically, it supposes a number of
iterative horn lengths adjustments in order to
modify own frequencies and achieve desired
frequency. Theoretically shortening lengths
are revealed in table 1.
The corespondance between the USH
frequency and shortening lenghts is
presented in table 1 and fig.9.
Visualization of results regarding the
sonotrode modal shapes and the relative
amplitude at resonance frequencyis are
presented in fig. 6, 7 and 8:
Fig. 6. The relative amplitude in case of a
2D axis symmetric model ( partial revolved)
Table 1. Theoretically shortening lengths
f [Hz]
Δ L1 [mm] Δ l2 [mm]
19750
19800
19850
19900
19950
20000
20050
20100
20150
20200
20250
Fig. 7. The relative amplitude for a 3D model
0
-0,12134
-0,24206
-0,36218
-0,48169
-0,60061
-0,71893
-0,83667
-0,95382
-1,07039
-1,18639
0
-0,12942
-0,2582
-0,38632
-0,5138
-0,64065
-0,76686
-0,89245
-1,01741
-1,14175
-1,26548
0.0
-0.2
[mm]
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
19
0
0
0
0
0
0
0
0
0
0
0
75 9 8 0 98 5 9 90 9 9 5 0 00 0 0 5 01 0 1 51 0 2 0 02 5
1
1
1
1
2
2
2
2
20
20
Variable
ΔL1 [mm]
Δl2 [mm]
Fig. 9. Shortening length vs. frequency
Fig. 8. Relative displacements of main points
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Nonconventional Technologies Review – no. 4/2011
Table 3. Diameters precision vs. frequency
ΔD1
ΔD2
D1
d2
f
Δf
[mm] [mm] [mm] [mm] [Hz]
[%]
52
0
40
0
19615
0%
Fig. 10. Shortening length position
51,9
52
51,9
-0,1
0
-0,1
40
39,9
39,9
0
-0,1
-0,1
19624 +0,04%
19606 -0,04%
19615
0%
52,1
52
52,1
+0,1
0
+0,1
40
40,1
40,1
0
+0,1
+0,1
19606 -0,04%
19624 +0,04%
19615
0%
52,1
+0,1
39,9
-0,1
19598
-0,08%
This results can be considered as a possible
method on adjusting resonant frequency of
ultrasonic horn.
Results on radius corner (fig.11) influence
on own frequency of the USH studied on a
3D shape model are presented in table 2.
Fig. 12. Stepped sonotrode using different
materials
Fig. 11. Corner radius
Another problem that can be solved by
this method is in case the body of the
sonotrode is made from one material and the
electrode-tool from another, in order to obtain
lower materials costs.
An example of such case used in
EDM+US is showed in figure 12, and the
corresponding materials properties are
presented in table 4 and 5.
Table 2. Radius vs. resonant frequency
R
frequency
Δf
[mm]
[Hz]
[%]
0
19686
0%
0,5
19642
-0,22%
1
19666
-0,10%
1,5
19694
+0,04%
2
19707
+0,10%
2,5
19735
+0,25%
3
19763
+0,39%
The junction to the smaller cross-section
should have a radius because there is a
danger of cracking here [2].
For sonotrodes with maximum crosssectional dimensions of 60 mm a R=10 mm
has been found to be satisfactory.
According to some authors [1] there is no
need of prescribing a specific radius, because
it’s close (or coincident) with the nodal plane
where the displacements are theoretically 0
or small enough, resulting in small stresses.
Resonant frequency variations of stepped
sonotrode with dimensional precision of
sonotrode
diameters
resulting
from
manufacturing are specified in table 3.
Table 4. Considered materials properties
ρ
E [Gpa]
Material
ν
[kg/m3]
OLC 45
0,31
210
7830
(Romanian steel)
Copper-99,5%
148
8800
Table 5. Sound velocity
Material
unit
OLC45
m/s
Cu – 99,5%
m/s
29
0,33
value
5125
4010
When sonotrodes which have already
been attuned are reworked by adjusting their
length frequency f0 becomes higher.
Afterwards it’s possible the sonotrode
frequency can quit the permissible tolerance
Nonconventional Technologies Review – no. 4/2011
interval and can not operated on a specific
transducer.
It is also possible to reduce it’s resonant
frequency again slightly-downwards, using
one of the methods presented in figure 12.
(a) – by shortening length L1 with Δl1;
(b) – by making a groove at the center of
gravity.
(a)
limitations of the FEM or because of
measurement errors for the real resonator.
From our previous studies, it was
established that the stability of EDM+US
finishing process was obtained increasing the
working gap through ignition voltage growing
and amplitude decrease of US oscillations.
FEM modelling of sonotrodes described
on this paper enables design time and
manufacturing costs to be effectively reduced
in practical EDM+US technological system
achievements.
Further studies must be carried on the
various geometrical shapes of sonotrodes as
one of the most important elements of the
ultrasonic assisting system. The main
dynamics characteristics (natural frequencies
and amplification factors) of sonotrode in the
resonant state should be studied according to
the geometric shape and dimensions.
(b)
ACKNOWLEDGEMENTS
Fig. 13. Methods for reducing sonotrode
resonant frequency [2],[5],[6]
Research
made
under
project
POSDRU/88/1.5/S/60370
co-funded
by
European Social Fund through the Sectoral
Operational Programme - Human Resources
Development 2007 - 2013.
5. CONCLUSIONS
It is very important to know the
resonance frequency and amplitude at
every point of the ultrasonic horn, because: it
allows precise determination of the nodal
plane position for locking in place of the
ultrasonic system for desired processing; the
parameters of the ultrasonic system can be
correlated with technological parameters,
especially at EDM+US, in order to avoid
short-circuits between electrode-tool and
workpiece; finite element method allows
testing of various shapes of ultrasonic horns
without the need to manufacture a real
prototype; it allows the correct selection of
the horn shape for a particular machining
process demands. It is also important the
selection of horn material for desired
amplitude of vibration.
If the FEM has been performed properly,
then the performance predicted by FEM
should agree reasonably with the real
sonotrode performance.
Agreement between modelling results
and real results should be especially good if
similar resonators have been previously
modeled and validated. Exact agreement
should not be expected, either because of
REFERENCES
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Prelucrarea prin eroziune cu unde ultrasonice –
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2. ZVEI - German Electrical Manufacturers
Association - Recommendations on methods,
construction and applications, online:
www.powerultrasonics.com/content/sonotrodedesign-and-manufacturing-instructions-zveihandbook / accessed - sept. 2011.
3. Nad, M., Ultrasonic horn design for
ultrasonic machining technologies, Applied and
Computational Mechanics 4, pp. 79–88, 2010.
4. Marinescu, N.I. et al., Solutions for
technological
performances
increasing
at
ultrasonic aided electrodischarge machining,
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ISBN 978-1-4200-7103-0, 2011.
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ULTRASONICS - Data, Equations and Their
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ISBN 978-0-8247-5830-1, 2009.
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